By using a center punch, you are putting a dimple in the screw head that will help you get the drill bit started. (You hit the punch with a hammer) By using a big enough drill bit you can take the head off and when the grate or whatever is removed there is usually enough of the screw left to turn out with a vise grips, if you want to reuse the holes.
You also may have hardened screw which makes it tougher, assuming you used sharp drill bits the first time, try a cobalt bit which is used for harder metals.